In mathematics and theoretical physics (especially twistor theory), twistor space is the complex vector space of solutions of the twistor. It was described in the 1960s by Roger Penrose and Malcolm MacCallum. According to Andrew Hodges, twistor space is useful for conceptualizing the way photons travel through space, using four complex numbers. He also posits that twistor space may aid in understanding the asymmetry of the weak nuclear force.

Informal motivation

In the (translated) words of Jacques Hadamard However, since there is no canonical way of doing so, instead all isomorphisms respecting orientation and metric between the two are considered. It turns out that complex projective 3-space. It turns out that vector bundles with self-dual connections(instantons) correspond bijectively to holomorphic vector bundles.

Formal definition

For Minkowski space, the solutions to the twistor equation are of the form

are two constant Weyl spinors are the Pauli matrices, and with a hermitian form

which is invariant under the group SU(2,2) which is a quadruple cover of the conformal group C(1,3) of compactified Minkowski spacetime.

Points in Minkowski space are related to subspaces of twistor space through the incidence relation

.

.

The geometric relation between projective twistor space and complexified compactified Minkowski space is the same as the relation between lines and two-planes in twistor space; more precisely, twistor space is

It has associated to it the double fibration of flag manifolds is the projective twistor space

is the compactified complexified Minkowski space

is

stands for projective space a Grassmannian a flag manifold. The double fibration gives rise to two correspondences (see also Penrose transform

by the Plücker embedding; the image is the Klein quadric.

See also

• Lorentz force

References